In friendly/matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

These notes are taken from a presentation by Duncan Murdoch, titled "Recent Developments in rgl".

The Singular Value Decomposition

For a square $n \times n$ matrix $A$, the SVD is $$ A = U D V^T $$ where

• $U$ and $V$ are $n \times n$ orthogonal matrices (i.e. $U^TU = V^TV = I$)
• $D$ is an $n \times n$ diagonal matrix with non-negative entries

Displaying a Matrix Graphically

• Matrices are representations of linear operators on vector spaces.
• The matrix $A$ is characterized by the behaviour of $y = A x$ as we vary $x$.
• Use the rgl package to develop a graphical representation of $3 \times 3$ matrices, e.g. $$ A = \left(\begin{array}{rrr} 1 & 0.1 & 0.1 \ 2 & 1 & 0.1 \ 0.1 & 0.1 & 0.5 \end{array}\right) $$

Attempt 1: display the actions on the basis vectors

A <- matrix(c(1,2,0.1, 0.1,1,0.1, 0.1,0.1,0.5), 3,3)
basis <- cbind(c(0,1,0,0,0,0), c(0,0,0,1,0,0), c(0,0,0,0,0,1))
segments3d(basis, lwd = 3)
segments3d(basis %*% t(A), col = "red", lwd = 5)
text3d(1.1*basis, texts = c("","x", "", "y", "", "z"), cex = 2)
text3d(1.1*basis %*% t(A), col = "red", 
       texts=c("", "x", "", "y", "", "z"), cex = 2)

Attempt 2: coloured ellipsoids.

sphere <- subdivision3d(cube3d(color=rep(rainbow(6),rep(4*4^4,6))),depth=4)
sphere$vb[4,] <- apply(sphere$vb[1:3,], 2, function(x) sqrt(sum(x^2)))
mult <- function(matrix, obj) transform3d(obj, t(matrix))
mat <- matrix(1:6, 2, 3)
layout3d(rbind(mat,mat+6), height = c(3,1,3,1), sharedMouse=TRUE)
shade3d(sphere)
next3d()
text3d(0,0,0, "Identity", cex=1.5)
next3d()
next3d(reuse=FALSE)
next3d(reuse=FALSE)
shade3d(mult(A, sphere))
next3d()
text3d(0,0,0,"A", cex=1.5)

svd <- svd(A)
U <- svd$u
D <- diag(svd$d)
V <- svd$v
next3d()
shade3d(mult(U, sphere))
next3d()
text3d(0,0,0, "U", cex=1.5)
next3d()
shade3d(mult(D, sphere))
next3d()
text3d(0,0,0, "D", cex=1.5)
next3d()
shade3d(mult(V, sphere))
next3d()
text3d(0,0,0, "V", cex=1.5)

friendly/matlib documentation built on Dec. 6, 2024, 9:25 a.m.